COUNTING FORMULA FOR SOLUTIONS OF DIAGONAL EQUATIONS

Abstract

Let N($d_1,...,{\;}d_n;c_1,...,{\;}c_n$) be the number of solutions $(x_1,...,{\;}x_n){\in}F^{n}_p$ of the diagonal equation $c_lx_1^{d_1}+c_2x_2^{d_2}+{\cdots}+c_nx_n^{d_n}{\;}={\;}0{\;}n{\geq},{\;}c_j{\;}{\in}{\;}F^{*}_q,{\;}j=1,2,...,{\;}n$ where $d_j{\;}>{\;}1{\;}and{\;}d_j{\;}$\mid${\;}q{\;}-{\;}1$ for all j = 1,2,..., n. In this paper, we find all n-tuples ($d_1,...,{\;}d_n$) such that the reduced form of ($d_1,...,{\;}d_n$) and N($d_1,...,{\;}d_n;c_1,...,{\;}c_n$) are the same as in the theorem obtained by Sun Qi [3]. Improving this, we also get an explicit formula for the number of solutions of the diagonal equation, unver a certain natural restriction on the exponents.

Keywords

• diagonal equations;
• finite fields

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References

1. J. Sichuan Univ. Natural Sci. v.3 On the Number of Solution of the Equation Σi=1 xi/di Ξ 0 (mod 1) and of Diagonal Equations in Finite Fields Granville, Shuguang Li;Sun Qi
2. Encylopedia Math. Appl. v.20 Finite Fields R. Lidl;H. Niederreiter
3. Finite Fields Appl. v.3 On Diagonal Equations over Finite Fields Sun Qi
4. Finite Fields Appl. v.2 On the Number of Solutions of Diagonal Equations over a Finite Field Sun Qi;Ping-Zhi Yuan
5. Proc. Amer. Math. Soc. v.103 Zeros of Diagonal Equations over Finite Fields Daqing Wan